Bar springs exhibit non-linear effects when subjected to high deflections. In a yaw rate sensor manufactured by the applicant (Robert Bosch GmbH model MM2R), a rotor is suspended in its center of rotation on an X spring that includes bar springs (see appended FIG. 3). This non-linear effect is clearly detectable even for deflections as small as ±4°. It is manifested in a shift of the resonance frequency of the system. In addition, areas exhibiting a plurality of stable operating states may appear as a result of the non-linearity. If a sudden transition occurs between the stable states during operation due to minor interference, for example, the performance of the yaw rate sensor may be considerably impaired.
In the yaw rate sensor (model MM2R) manufactured by the present Applicant and schematically illustrated in FIG. 3, due to a centrally mounted X spring 2, the bar spring element has a centrally mounted and centrally symmetric inertial mass 1 which is excited to a periodic rotary oscillation in a lateral plane x, y having positive and negative deflections i, a of the same magnitude about a rest position O (angle γ designates the deflection angle in positive direction i) by symmetrically acting comb drives. A first comb drive pair has two comb drive units KI1 and KA1 opposite one another, which act upon a first drive point P1 situated on a circular arc-shaped peripheral segment of inertial mass 1. These two first comb drive units KI1 and KA1, which are provided for a deflection in positive direction i and negative direction a, respectively, are situated parallel to a an imaginary straight line connecting the center of X spring 2 to the rest position of inertial mass 1. A second pair of comb drive units KI2, KA2 for positive deflections i and negative deflections a, respectively, situated centrally symmetrically to the first pair of comb drive units KI1, KA, acts upon a second point of application P2 diametrically opposite first point of application P1 on inertial mass 1. The latter two comb drive units KI2 and KA2 are situated parallel to an imaginary straight line connecting the center of inertial mass 1 to point O denoting the rest position.
A control unit 3 generates drive voltages UPKI,drive and UPKA,drive for comb drive units KI1 and KI2 (for excitation in positive deflection direction i) and for comb drive units KA1 and KA2 (for excitation in negative deflection direction a), respectively, as shown in appended FIG. 4. FIG. 4 shows in top part A the periodic and, in the ideal case, harmonic excitation function γ(t) of inertial mass 1 in positive direction i and negative direction a, which has the period 2π.
In center FIG. 4B, a dashed line shows the square pulse-shaped drive voltage UPKA,drive for comb drive units KA1 and KA2 for deflection in negative direction a, while bottom FIG. 4C shows drive voltage UPKI,drive for comb drive units KI1, KI2 for positive deflection in phase opposition to drive voltage UPKA,drive, the positive deflection also having square pulses having the periodicity of periodic excitation oscillation γ(t) shown in FIG. 4A.
In addition, FIG. 4 shows that the pulses of drive voltage UPKI,drive for positive deflection i according to FIG. 4C are generated symmetrically to the positive zero crossings of periodic oscillation γ(t) of inertial mass 1, while the pulses of drive voltage UPKA,drive for negative deflection i according to FIG. 4B are generated symmetrically to the negative zero crossings of periodic excitation oscillation γ(t) according to FIG. 4A. Since the comb drive units (KI and KA) schematically shown in FIG. 3 are each able to exert a force in one direction only, a plurality of comb drive units are needed to set the rotor in an oscillating motion.
For practical reasons, a square-wave voltage is generated for the drive voltage applied by control unit 3 instead of the sinus curve. This is accomplishable using a control logic and a voltage pump in control unit 3. In principle, a voltage pump is made up of a capacitor which is charged. This allows higher voltages than the operating voltage to be generated for a short period. This voltage is used if needed.
The non-linear bending of the bar is describable using Duffing's differential equation. Since this is a known differential equation, a detailed analysis of the dynamic properties is not necessary. Instead, the two main effects (frequency shift and instability) are briefly explained with reference to appended FIGS. 1 and 2. The mechanical non-linearity of the X spring may be described using Duffing's differential equationMdrive=Jz·γ+bt,z·γ+ktz,0·(1+ktz,NLγ2)γ  (1)orMdrive=Jz·γ+bt,z·γ+ktz,0·γ+ktz,0ktz,NLγ3  (2)
The parameters are determined via the appropriate finite-element computations.
As can be seen from equation (2), this is a second-order differential equation for oscillation, having linear attenuation term bt,z (velocity-proportional attenuation). γ denotes the angle describing the deflection of the rotor. The only difference with respect to the “standard differential equation for oscillation” is additional non-linear term ktz,0ktz,NLγ3. Term ktz,0 is the linear spring constant (torsion). Term ktz,NL describes the non-linearity. Jz denotes the moment of inertia of the rotor about the z axis. The rotor oscillation is excited via drive moment Mdrive.
Appended FIG. 1 shows the resonance curve of the gain in the linear case with attenuation. The gain factor attains its maximum at the exact moment when the system is excited by its intrinsic frequency. In the non-linear case, the maximum of the resonance curve is shifted to the right. As can be seen from appended FIG. 2, the resonance curve tends additionally to the right. The one-to-one correspondence between gain and excitation frequency is thus lost. The gain in each individual case depends on the previous history. For example, in the case of a smooth increase in the excitation frequency, there is a slow transition from area I to area II and thus to point 1 in FIG. 2. In the case of supercritical excitation and a slower reduction in the drive frequency, the operating state migrates from area III to point 2 in area II. This makes two different states possible for one drive frequency. In general, this is an undesirable effect, because, for example, interference pulses may cause a sudden transition from point 1 to point 2. The risk of a sudden change exists in the entire area II (shaded in FIG. 2).
Only the basic analytical equation in principle is given here as the operating principle of comb drives:Fx≈ε0ΔU2(h/d0)  (3)
The following relationships are discernible from equation 3:                The greater voltage difference (ΔU) on the capacitor plates of the comb drive, the greater drive force Fx.        The greater height h of the comb drive structure, the greater drive force Fx.        The smaller distance d0 between the capacitor plates of the comb drive, the greater drive force Fx.        
These are only the basic relationships. Detailed examination shows that the force is caused by the formation of stray fields. Therefore, in a more accurate analysis, the field lines must be used for computing the force. For more complex geometries, the relationships may be determined using finite element computations.